sanju thesis
TRANSCRIPT
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Optimal Allocation of Series CapacitorThesis submitted in partial fulfillment of the requirements for the award of
degree of
Master of Engineeringin
Power Systems & Electric Drives
Thapar University, Patiala
By:Sanju Bala(80641020)
Under the supervision of:Mr. Parag NijhawanSr. Lecturer, EIED
JUNE 2008
ELECTRICAL & INSTRUMENTATION ENGINEERING DEPARTMENTTHAPAR UNIVERSITY
PATIALA – 147004
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Acknowledgement
Words are often less to reveal one’s deep regards. An understanding of work like this is
never the outcome of the efforts of a single person. I take this opportunity to express my
profound sense of gratitude and respect to all those who helped me through the duration
of this thesis.
First of all I would like to thank the Almighty, who has always guided me to work
on the right path of the life. My greatest thanks are to my parents who bestowed ability
and strength in me to complete this work.
This work would not have been possible without the encouragement and able
guidance of my supervisor Mr. Parag Nijhawan. His enthusiasm and optimism made
this experience both rewarding and enjoyable. Most of the novel ideas and solutions
found in this thesis are the result of our numerous stimulating discussions.
I take this opportunity to express my gratitude and sincere thanks to Dr. Samarjit
Ghosh, Prof. & Head, Electrical & Instrumentation Engineering Department for
their valuable suggestion and for providing me the opportunity to complete my thesis
work simultaneously.
I am most indebted to the P.G. coordinator Dr. Sanjay Kumar Jain for his help,
assistance and suggestions during my work.
I am also very thankful to the entire faculty and staff members of Electrical and
Instrumentation Engineering Department for their direct–indirect help, cooperation, love
and affection, which made my stay at Thapar University memorable.
I wish to thank all my classmates for their time to time suggestions and
cooperation without which I would not have been able to complete my work.
(SANJU BALA)
(80641020)
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Abstract
In recent years, energy, environment, right-of-way and cost problems have
delayed the construction of both generation facilities and new transmission lines. These
problems have necessitated a change in the traditional concepts and practices of power
systems. Better utilization of existing power system capacities by installing compensating
devices has become imperative.
Series capacitors can be utilized to change the reactive power flow through the
line. Thus the power transfer capabilities will be increased and the transmission
investment cost will be reduced. This compensation scheme is one facet of FACTS that
has been widely accepted as a solution for the limitations created by overstretched
generation and transmission systems.
The objective of this study is to find the optimal location of compensation
devices. This will lead to the better utilization of the existing infrastructure like
transmission lines. Parameters like sensitivity analysis, line flow index etc. are calculated
and analyzed to find the best location for the compensation device.
Some computational tools are needed in assisting the analysis of the effects on
the system performance due to the control settings and the new devices. The existing
system analysis tool is Optimal Power Flow (OPF). Thus, it is necessary to extend the
existing traditional OPF to include the representation of variable series capacitors. Since
the main concern in this study is to compensate reactive power flows and then optimize
the system using Linear Programming. In this thesis work IEEE 30 bus test system is
considered.
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Table of Contents
Certificate ……………………………………………………………………………….. i
Acknowledgement ……………………………………………………….... …………… ii
Abstract …………………………………………………………………………………. iii
Table of Contents ………………………………………………………….……………. iv
Chapter 1 Introduction …………………………………………………………………... 1
1.1 Overview ……………………………………………………..................... 1
1.2 Literature Review ……………………………………………………….... 2
1.3 Scope of Work ………………………………………………...…………. 8
1.4 Organization of Thesis ………………………………………………….... 8
Chapter 2 Reactive Power Compensation ………………………………………………. 9
2.1 Introduction ………………………………………………………………... 9
2.2 Shunt Compensation .……………………………………………………… 9
2.3 Series Compensation ……………………………………………………….10
2.4 Synchronous Condensers …………………………………………………..10
2.5 Effects of Reactive Power Compensation ………………………………….11
2.5.1 Reduced Power System Losses ………………………………………11
2.5.2 Release of Power System Capacity ………………………………… 11
2.6 Drawbacks of Reactive Power Compensation ……………………………..11
Chapter 3 Optimal Power Flow …………………………………………………………13
3.1 Introduction ………………………………………………………………..13
3.2 OPF Problem Formulation …………………………………………………14
3.3 Linear Programming Based Optimal Power Flow …………………………15
3.3.1 Algorithm …………………………………………………………….18
Chapter 4 Selection for the Proper Location of Series Capacitor ..……………………...19
4.1 Introduction ………………………………………………………………...19
4.2 Various Parameters ………………………………………………………...20
4.2.1 Sensitivity Coefficients ………………………………………………. 20
4.2.2 Line Flow Index ……………………………………………………… 22
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4.2.3 Voltage Drop …………………………………………………………. 22
4.3 Results and Discussion …………………………………………………… 22
Chapter 5 Conclusion and Future Work ……………………………………………….. 27
5.1 Conclusion ………………………………………………………………..27
5.2 Future Work ………………………………………………………………27
References ……………………………………………………………………………… 28
Appendix I ………………………………………………………………………………31
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Chapter 1
Introduction
1.1 OverviewThe transmission of electricity differs from transportation of any typical
commodity by some inherent aspects such as: the production needs to match the
consumption at the same time; system control is not an easy task; the electricity flows
do not usually follow the economic law. The last aspect is normally observed when
transmission systems are included in an economic dispatch problem. One way to
minimize the operational costs caused by an overloaded transmission system is
through the installation of Flexible AC Transmission System (FACTS) devices in the
system or some reactive power compensating device like Series Capacitor. They are
able to change power flows by modifying the network parameters. The optimal
allocation and utilization of compensating devices are important issues primarily due
to their cost. Recently network blackouts related to voltage collapse tend to occur
from lack of reactive power support in heavily stressed conditions, which are usually
triggered by system faults. Power electronics based equipment, or Flexible AC
Transmission Systems (FACTS), provide proven technical solutions to active and
reactive power control and voltage stability problems.
The FACTS is a concept based on power electronic controllers, which enhance
the value of transmission networks by increasing the use of their capacity. As these
controllers operate very fast, they enlarge the safe operating limits of a transmission
system without risking stability. Series Capacitor, Shunt Inductor etc. are the
backbone of FACTS devices. The best or optimal location of the Series Capacitor is
tried to be obtained. The same idea can be extended to find out the optimal location of
FACTS device.
These devices make the application of a large amount of Var
compensation more efficient, flexible and attractive. Consequently, a series of
questions have been raised frequently by utility planners and manufacturers: Where is
the right location and what is the right size for the installation of reactive power
compensators considering technical and economic needs? Can the models, methods,
and tools used for static Var planning be applied in dynamic Var planning? The
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answers to these questions are needed for utilities to make better use of these new
power electronic controlled Var sources. In order to answer the above questions, it
should be stated that optimal allocation of static and dynamic Var sources belongs to
the Reactive Power Planning (RPP) or Var planning category. RPP deals with the
decision on new Var source location and size to cover normal, as well as, contingency
conditions. The planning process aims at providing the system with efficient Var
compensation to enable the system to be operated under a correct balance between
security and economic concerns. This report focuses on the ability of compensating
devices mainly Series Capacitor to change the overall losses of the system and its
allocation in the power system, then system is optimized using Linear Programming.
1.2 Literature ReviewSeveral research papers and reports addressed the subject of optimal capacitor
placement in distribution systems. The followings present a brief review of the work
undertaken so far.
Kaplan presented a computerized trial and error heuristic method for
optimizing the present worth of revenue savings [17]. The savings are associated with
released system capacity and the energy loss reductions. Both fixed and switched
capacitor banks and their installation cost were also considered. The availability of the
capacitor banks in accordance with the standards and released capacity cost were
included as well.
The optimization process consists of three major steps. First is the choice of
the location and type for the smallest standard size bank. The program scans all feeder
branches moving along the branch toward the substation. Second is the improvement
of the solution considering the standard bank size. The aim of this step is to increase
the objective function. Third is the selection of the type of control for switched banks.
The method, however, does not consider system voltage limitations or the economic
effect of voltage rise resulting from capacitor applications.
S. Rama Iyer et al. defined the objective function of the optimal capacitor
placement problem as a maximization of revenue savings resulting from power loss
reduction [25]. The objective function is solved by a mixed-integer programming
technique subjected to the following constraints:
1. Upper and lower limits of generator voltage magnitudes.
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2. Upper and lower values for the transformer tap settings.
3. Upper limit on the number of units in each capacitor bank due to technical reasons
such as switching voltage surges.
The proposed method finds the optimal location of a capacitor by the coordinated
variation of generator voltages, transformer tap settings and the number of units in
each capacitor bank. The method has some features which result in considerable
savings in computer time and memory. First is the elimination of dependent variables
in the problem formulation. Second is the decomposition of the problem into two
smaller sub problems. Third is the avoidance of the load flow calculation.
A. A. El-Kib et al. presented a new model for radial distribution systems in
[1]. The model considers asymmetrical and multi-grounded feeders. They also supply
unbalanced loads. Based on the model, the optimal size, locations and switching
intervals of fixed and switched capacitors are determined. The paper, however, did
not discuss optimal number of fixed and/or switched capacitors to achieve the
maximum savings. Capacitor sizes are treated as continuous variables which is not the
case in real-life where capacitor sizes are discrete. A numerical example is presented
for a real multi-grounded, three-phase feeder with lateral branches and five wire sizes.
S. Ertem and J. Tudor presented an objective function representing monetary
savings that result from capacitor allocation in terms of system voltages and angles
and the reactive power to be allocated [26]. The constraints are the maximum and
minimum system voltages. The method takes the load uncertainty into account. This
is done to prevent both over-compensation and the under-compensation. To reduce
system variables and to avoid the numerical instability, sensitivity analysis and pattern
recognition techniques are used. The proposed solution is based on non-linear
programming technique where two models are presented. The first one solves the
problem subject to approximate constraints. The output of this model is used as a
starting solution to the second model. The second model has the exact formulation of
the constraints using the method of approximate programming.
R. Rinker and D. Rembert presented a method to optimally allocate capacitors
along a distribution line using the data gathered by reactive current recorders installed
at major feeder taps [23]. The data is used later to achieve maximum savings by
placing the proper sizes of capacitors at the appropriate places. The method assumes
that the current readings recorded at a node are applicable to the entire section which
follows the node. It places the capacitors in different types and sizes in a systematic
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trial and error procedure. It can also be improved if the program is extended to handle
split feeders or nodes placed on significant branches. M. Baran and F. Wu presented a
nonlinear mixed integer programming formulation to solve the capacitor placement
problem [19]. The objective function to be maximized is the revenue savings resulting
from energy loss reduction minus the capacitors installation costs. Voltage constraints
and load variations are considered as the problem constraints. The solution
methodology decomposes the problem into a master and a slave subproblem. In the
master sub-problem an integer programming is used to find the optimal location and
the number of capacitor banks to be placed. The slave sub-problem is used by the
master sub-problem to find the capacitor type and settings. The proposed solution
model was tested by considering two different test systems. Although voltage
regulators were not considered in this paper, the method can be extended to account
for their presence.
Y. Baghzoz presented an exhaustive search method to solve for the optimal
solution of the capacitor placement problem and nonlinear load models are
incorporated in the problem formulation [30]. The paper assumes that the system is
balanced and that all loads vary in a conforming way. The active and reactive powers
represent the fundamental frequency quantities. The loads are partitioned into linear
and non-linear loads. The nonlinear loads are assumed to have the same displacement
factor. To show the effect of the load model on the final solution, the problem
considers three different load models. These are the base case model, the most
accurate model and the constant impedance model. The problem constraints are the
maximum number of capacitors assigned at a particular location. They also include
the maximum number of capacitors on the entire feeder. The maximum and minimum
r.m.s. voltage at a certain bus, the maximum peak value and the total harmonic
distortion are also considered. Numerical results show that consideration of load non-
linearity substantially changes the optimal solution of the problem; the test system
used by the author in this paper is a small one with 9 buses only. Exhaustive search,
however, shows that it is not possible to be used for large systems.
S. Tripathy and M. Haridas discussed an analytical method for the problem of
installing shunt capacitors on distribution feeders [27]. The paper gives a detailed
derivation for the loss reduction in a distribution feeder. Based on this, the optimal
size and locations of shunt capacitors are calculated. The paper also discusses the use
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of the normalization technique in case of different conductor sizes and non-uniformly
distributed reactive loads.
C. Chen et al. presented a computer simulation program to solve the optimal
capacitor placement problem by means of non-linear programming while considering
the mutual coupling among phase conductors [5]. Mutual coupling is determined by
the types of conductors, horizontal and vertical distance among conductors. The
objective function to be minimized consists of peak power losses, energy losses and
capacitor installation cost. It is assumed that the installation cost/kVar is constant.
Also, the capacitor values are treated as discrete variables. The minimum and
maximum bus voltages are taken as the system constraints. A 34-bus unbalanced and
10-bus balanced test system was considered. Simulation results revealed that the
inclusion of mutual coupling is very important. If ignored, serious overcompensation
and extra power losses may result. In addition, the effect of mutual coupling is more
important for unbalanced systems.
Bala et al. presented a sensitivity-based optimization method to optimally
allocate shunt capacitors among a radial distribution feeder [11]. The solution should
find optimal size, type and location of the capacitors. Several researchers attempted to
solve the capacitor placement problem by using different heuristic and iterative
methods. These methods provide reasonable solutions with minimum complexity.
Chiang et al. solved the general capacitor placement problem in a distribution system
by means of the simulated annealing method [7,8]. The objective of this work is to
maximize the energy loss reductions in the system while considering the capacitors
installation costs. The capacitor cost function is step-like and non-differentiable. The
constraints are the upper and lower voltage limits and the load variations.
Chiang et al. presented a more comprehensive methodology based on
simulated annealing mixed with greedy search to solve the general capacitor
placement problem [9,10]. The purpose of this mix is to get a good-quality solution in
a shorter time. The proposed method can be applied to large scale unbalanced, radial
or loop distribution network. Accurate mathematical modeling of distribution lines,
shunt capacitors, transformers, loads and co-generators were employed. The aim is to
find the optimal location to install (or replace or remove) capacitors, types and sizes
during each load level and the control schemes for each capacitor. The objective
function takes into consideration the installation (or replacement or removal) cost of
capacitors and the system energy losses. The constraints of the problem are load flow,
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line flow capacity and voltage magnitude constraints. The cost function of capacitors
is a step-like and hence a non-differentiable function. Capacitor sizes and control
settings are treated as discrete variables.
Sundhararajan and Pahwa solved the general capacitor placement problem in a
distribution system using a genetic algorithm [28]. Genetic Algorithm, as an
optimization technique can reach a near optimal solution in a lesser time than that of
the simulated annealing. The objective function presented is not constrained.
However, if a constraint is to be incorporated, then a penalty function should be
included. Sensitivity analysis is used in this paper as an aid for the Genetic Algorithm
to help select the candidate locations of capacitors. Top two or three buses in each
lateral branch are chosen as candidate locations. A 9-bus and a 30-bus test systems
were used to check the robustness of the method. Yann-Chang Huang et al.
introduced a Tabu Search-based method to solve the capacitor placement problem
[32]. To start with, heuristic and engineering judgments are used to select the
potential locations where capacitors can be installed. Then, a sensitivity analysis is
used to determine the candidate locations. In comparison with simulated annealing,
the authors concluded that Tabu Search method gives the same results with shorter
computing time.
Z. Wu proposed to solve the capacitor placement problem by means of
Maximum Sensitivity Selection (MSS) method [33]. The MSS decomposes the
problem into sub- problems to make the solution simpler. The objective function
considers the peak power loss, energy losses and the installation cost of capacitors.
The problem constraints are system minimum and maximum voltages and total
harmonic distortion. The paper assumes that all loads change in a conforming way,
load variations can be approximated by discrete levels, loads are linear and balanced,
skin effect of higher harmonics is neglected and the substation is the only harmonic
source. Karen Nan Miii et al. proposed to solve the general capacitor placement
problem by a Genetic Algorithm followed by a sensitivity-based heuristic method
[16]. Genetic algorithm is employed to find a high quality solution that is used as an
initial guess for the sensitivity-based heuristic. The objective function includes the
cost of placement or replacement of capacitor banks and the cost of real power loss.
The problem constraints are the power flow constraints, operational constraints on bus
voltages and the line flow ratings. Simulation results revealed that the proposed
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hybrid algorithm outperformed the Genetic Algorithm alone and the sensitivity
heuristic alone in tents of both speed and quality.
M. Baran and F. Wu solved the optimal sizing problem of capacitors placed on
a radial distribution system as a special case of the general capacitor placement
problem [18].The sizing problem is formulated as a nonlinear programming problem.
The formulation incorporates the ac power flow model for the system and the voltage
constraints.
Y. Baghzouz and S. Ertem solved the optimal sizing of distribution capacitors
as a special case of the general capacitor placement problem using a simple heuristic
method based on the method of local variations [31]. Potential harmonic iterations
like resonance conditions, high harmonic distortion factor and additional harmonic
power losses are considered in the problem formulation. The savings associated with
power loss reduction may have to be sacrificed to control the total harmonic distortion
to acceptable limits. The paper assumes that the line capacitance are negligible,
system is balanced, all loads are linear and time invariant, harmonic generation is only
from the substation voltage supply, only fixed capacitors are used and capacitors are
represented as constant admittances.
J. Grainger et al. presented a generalized procedure based on non-linear
programming for optimizing the net savings resulting from power loss reduction
caused by capacitor installation [12]. The system unbalance is taken into account and
mutual coupling between the phases is considered. The paper, however, assumes only
fixed capacitors are to be installed. Moreover, capacitor sizes are treated as
continuous variables. Also, capacitor installation cost is neglected. Jin-Chang et al.
decouple the general capacitor placement problem into two sub- problems: the
capacitor placement sub-problem and the real-time control sub-problem [13,14]. A
quadratic integer programming based approach is proposed for the capacitor
placement sub-problem to determine the number, locations and sizes of capacitors to
be placed. The real-time control sub-problem is formulated as another quadratic
integer programming problem to determine the control settings for the different
loading conditions [2].
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1.3 Scope of WorkFrom the literature review, it is observed that the work on the investigation on
power with compensating devices is very much diversified. However it is observed
that there is a scope to investigate the effectiveness of compensating devices in power
flow solution without violating security constraints. And to find the location of these
devices in Power System.
The objective of the proposed work is to study the optimal power flow
solution with Series Capacitor and to devise the strategy for the allocation of Series
Capacitor to improve the optimal power flow solution. The proposed work includes
optimal allocation of Series Capacitor and then system is optimized using Linear
Programming.
1.4 Organization of ThesisChapter 1 describes introduction to the problem. It presents the literature review on
compensating devices and their allocation in power system to minimize losses.
Chapter 2 covers reactive power compensation. It presents a brief idea about series,
shunt compensation etc. and its effects.
Chapter 3 describes Optimal Power Flow problem formulation and explains Linear
Programming based optimization technique.
Chapter 4 presents the methodology to allocate Series Capacitor in power system to
improve system performance. Various parameters like sensitivity coefficients, line
flow index etc. are calculated to allocate the device.
Chapter 5 presents the conclusion of this thesis work and discusses the future scope of
the work.
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Chapter 2
Reactive Power Compensation
2.1 IntroductionReactive power compensation is an important issue in electric power systems,
involving operational, economical and quality of service aspects. Consumer loads
(residential, commercial and industrial sectors) impose real and reactive power
demand, depending on their characteristics. Real power is converted into “useful”
energy, such as light or heat. Reactive power must be compensated to guarantee an
efficient delivery of real power to loads [4].
The reactive power is essential for creating the needed coupling fields for
energy devices. It constitutes voltage and current loading of circuits but does not
result in average (active) power consumption and is an important component in all ac
power networks. Reactive power control for a line is often called Reactive Power
Compensation. External devices or subsystems that control reactive power on
transmission lines are known as Compensators. The objectives of line compensation
are
· To increase the power transmission capacity of the line, and/or
· To keep the voltage profile of the line along its length within acceptable
bounds to ensure the quality of supply to the connected customers.
Because reactive power compensation influences the power transmission capacity of
the connected line, controlled compensation can be used to improve the system
stability by changing the maximum power- transmission capacity.
2.2 Shunt CompensationWhen fixed inductors and/or capacitors are employed to absorb or generate
reactive power, they constitute passive control. Shunt devices may be connected
permanently or through a switch. Shunt reactors compensate for the line capacitance,
and because they control over voltages at no loads and light loads, they are often
connected permanently to the line, not to the bus. Shunt capacitors are used to
increase the power transfer capacity and to compensate for the reactive voltage drop
in the line. The applications of shunt capacitors require careful system design. The
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circuit breakers connecting shunt capacitors should withstand high-charging in-rush
currents. The addition of shunt capacitors creates higher frequency resonant circuits
and can therefore lead to harmonic over voltages on some system buses.
2.3 Series CompensationSeries capacitors are used to partially offset the effects of the series
inductances of the lines. Series compensation results in the improvement of the
maximum power-transmission capacity of the line. The reactive power absorption of a
line depends on the transmission current, so when series capacitors are employed,
automatically the resulting reactive power compensation is adjusted proportionately.
Also, because the series compensation effectively reduces the overall line reactance, it
is expected that the net line-voltage drop would become less susceptible to the loading
conditions.
In an interconnected network of power lines that provides several parallel
paths, for power flow between two locations, it is the series compensation of a
selected line that makes it the principal power carrier. A practical upper limit of series
compensation may be as high as 0.75 pu.
2.4 Synchronous CondensersSynchronous condensers or synchronous compensators were the only fully
controllable reactive power devices available for power systems until mid 1970s. A
synchronous condenser is a synchronous machine, the reactive power output of which
can be continuously controlled by varying its excitation current. When the
synchronous machine is connected to the ac system and is under excited, it behaves
like an inductor, absorbing reactive power from the ac system. However, when it is
overexcited, it functions like a capacitor, injecting reactive power into the ac system.
The machine is normally excited at the base current when its generated voltage equals
the system voltage; it thus floats without exchanging reactive power with the system.
A synchronous condenser is usually connected to the EHV ac system through a
coupling transformer [24].
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2.5 Effects of Reactive Power Compensation2.5.1 Reduced Power System Losses
The reduction in power system losses, due to the installation of capacitor
banks, can result in an annual gross return of as much as 15 percent on the capacitor
investment. Although it is seldom sufficient to justify the installation of capacitor
banks on the economical benefits of power loss reduction alone, it is certainly an
added benefit [8]. In most industrial plant power distribution systems, the power
losses vary from 2.5 to 7.5 percent of the load kWh. This depends upon hours of full
load and no-load plant operation, wire size, and length of the main and branch feeder
circuits. Capacitors are effective in reducing only that portion of the losses that is due
to the kVars current. Losses are proportional to the current squared, and since current
is reduced in direct proportion to power factor improvement, the losses are inversely
proportional to the square of the power factor. Hence, as power factor is increased
with the addition of capacitor banks to the system, the magnitude of the losses are
reduced. The connected capacitor banks have losses, but they are relatively small.
Losses account for approximately one third of one percent of the kVar rating.
2.5.2 Release of Power System Capacity
When capacitors are placed in a power system, they deliver kVars. This
introduces furnishing magnetizing current for motors, transformers and other similar
plant, thus reducing the current from the power supply. Less current means less KVA
or load placed on the transformers and main branch feeder circuits. This means
capacitors can be used to reduce overloading or permit additional load to be added to
existing feeders. Release of system capacity by power factor improvement and
especially with capacitors is becoming extremely important due to the associated
economic and system benefits.
2.6 Drawbacks of Reactive Power CompensationOne of the main drawbacks that must be considered when introducing capacitor
banks into a power system includes resonance. Resonance is a condition whereby the
capacitive reactance of a system offsets its inductive reactance. This causes the
resistive elements as the only means of impedance in the system. The frequency at
which this offsetting effect takes place is called the resonant frequency of the system.
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In the event of a harmonic generating source near or at the resonant frequency,
dangerous amplification to voltages and currents will occur. In turn this can cause
damage to capacitor banks and other electrical equipment connected to the system.
Resonance can take the form of either series or parallel type. This depends on how the
reactive elements are arranged throughout the system.
As discussed previously, a high power factor makes better use of the available
capacity of the power system. However, the occurrence of harmonics may create
parallel and series resonance conditions, causing destructive consequences to the
system. From the perspective of harmonic sources, shunt capacitor banks appear to be
in parallel with the systems short circuit reactance. For ideal circuit elements, and
neglecting saturation and other non-linear effects, inductive reactance increases
directly as frequency increases while capacitive reactance decreases as frequency
increases.
At the resonant frequency of the system, the parallel combination of the
capacitor bank and the source reactance appears as large impedance. If this frequency
happens to match with one generated by the harmonic source, then dangerous
voltages and currents will increase disproportionately, causing damage to capacitors
and other electrical equipment. Capacitors are also known to fail due to harmonic
overload. Since the impedance of a capacitor is inversely proportional to frequency,
harmonic currents may load capacitors beyond their limit causing them to fail [4].
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Chapter 3
Optimal Power Flow
3.1 IntroductionMathematical optimization (algorithmic) methods have been used over
the years for many power systems planning, operation, and control problems.
Mathematical formulations of real-world problems are derived under certain
assumptions and even with these assumptions; the solution of large-scale power
systems is not simple due to the uncertainties in power system problems. It is
desirable that solution of power system problems should be optimum globally, but
solution searched by mathematical optimization is normally optimum locally. These
facts make it difficult to deal effectively with many power system problems through
strict mathematical formulation alone.
Optimal power flow was first discussed by Carpentier in 1962 and took a long
time to become a successful algorithm that could be applied in everyday use. The
OPF constitutes a static nonlinear optimization problem which computes optimal
settings for electrical variables in a power network, for given settings of loads and
system parameters.
Economic dispatch, however, only considers rea1 power generations and
represents the electrical network by a single equality constraint, the power balance
equation. The addition of losses to the network model, starting in 1943, resulted in the
classical economic dispatch. The optimality conditions for these methods constitute
the equal incremental cost criterion (EICC). These methods are simple and fast
because they limit the network modeling to its simplest expression. Equal incremental
cost methods constitute the most popular economic dispatch tool.
Optimal power flow methods satisfy the nonlinear constraints using an
iterative scheme where iteration is based on some simpler linear subproblem.
Nonlinear programming methods include successive approximation, gradient and
successive linear programming. Linear programming method include simplex method,
artificial variable method etc.. In this thesis IEEE 30 bus test system is optimized
using Linear Programming [20].
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3.2 OPF Problem Formulation The objective function to be minimized is given below:
2
( )
( )
i ii
i i i i i i
f F Pg
whereF Pg a P b P c
=
= + +
å
This is the sum of operating cost over all controllable power sources.
( )i iF Pg = Generation cost function for Pgi generation at bus i.
The cost is optimized with the following constraints.
· The inequality constraint on real power generation at bus imin maxi i iPg Pg Pg£ £
wheremin max.i iPg and Pg are respectively minimum and maximum values of real
power generation allowed at generator bus i.
· The power flow equation of the power network
( , ) 0g V j =
where
( , )( , ) ( , )
( , )
neti i
neti i
netm m
P V Pg V Q V Q
P V P
j
j j
j
ì -ï
= -íï -î
where
Pi and Qi are respectively calculated real and reactive power for PQ bus i.
Pinet and Qinet are respectively specified real and reactive power for PQ bus i.
Pm and Pmnet are respectively calculated and specified real power for PV bus m.
V and φ are voltage magnitude and phase angles at different buses.
· The inequality constraint on reactive power generation Qgi at each PV busmin maxi i iQg Qg Qg£ £
where Qgimin and Qgi
max are respectively minimum and maximum value of
reactive power at PV bus i.
· The inequality constraint on voltage magnitude V of each PQ busmin max
i i iV V V£ £
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where Vimin and Vi
max are respectively minimum and maximum voltage at bus i.
· The inequality constraint on phase angle φi of voltage at all the buses imin maxi i ij j j£ £
where φimin and φi
max are respectively minimum and maximum voltage angles
allowed at bus i.
· MVA flow limit on transmission linemax
ij ijMVAf MVAf£
where MVAfijmax is the maximum rating of transmission line connecting bus i
and j
3.3 Linear Programming Based Optimal Power FlowTo obtain LPOPF firstly power flow equations are solved. The power
flow equations can be for the DC representation, the decoupled set of AC equations,
or the full AC power flow equations. The choice will affect the difficulty of obtaining
the linearized sensitivity coefficients and the convergence test used. Firstly nonlinear
input-output or cost functions (Figure 3.1) are expressed as a set of linear functions.
Non-linear function can be approximated as a series of straight-line segments as
shown in Figure 3.2. The three segments are represented as 1, 2, 3i i iP P P and each
segment has a slope designated: si1, si2, si3 for ith generator.
Then the cost function itself ismin
1 1 2 2 3 3( ) ( )i i i i i i i i i iF P F P s P s P s P= + + +
and 0≤Pik≤Pikmax for kth segment
and Pi=Pimin+Pi1+Pi2+Pi3
The cost function is now made up of a linear expression in the Pik values.
- 16 -
Figure 3.1: A Non-Linear Cost Function Characteristic.
Figure 3.2: A Linearized Cost Function
In the formulation of the optimal power flow using linear programming, we have
control variables (like generator output and generator bus voltage) in the problem. We
do not consider to place the state variables into LP, nor all the power flow equations.
Constraints are set up in the LP that reflects the influence of changes in the control
variables only. The control variables are represented as the ‘u’ variable. The
constraint that is considered in an LPOPF is the constraint that represents the power
balance between real and reactive power generated and consumed in the load and
losses.
Fi
Pi
Fi
Pi Pimin Pi1 Pi2 Pi3
- 17 -
The real power balance equation is:
Pgen - Pload - Plosss=0
The loss term represents the I2R losses in the transmission lines and transformers. We
can take derivatives with respect to the control variables, u, and this result in:
0gen load loss
u u u
P P Pu u uu u u
¶æ ö ¶ ¶æ ö æ öD - D - D =ç ÷ ç ÷ ç ÷¶ ¶ ¶è ø è øè øå å å
If we make the following substitution:0u u uD = -
then, the power balance equation becomes
gen load lossp
u u u
P P Pu u u Ku u u
¶æ ö ¶ ¶æ ö æ ö- - =ç ÷ ç ÷ ç ÷¶ ¶ ¶è ø è øè øå å å
where
0 0 0gen load lossp
u u u
P P PK u u uu u u
¶ ¶ ¶= - -
¶ ¶ ¶å å å
A similar equation can be written for the reactive power balance:
0gen load loss
u u u
Q Q Qu u uu u u
¶æ ö ¶ ¶æ ö æ öD - D - D =ç ÷ ç ÷ ç ÷¶ ¶ ¶è ø è øè øå å å
where the loss term is understood to include I2X as well as the charging from line
capacitors and shunt reactors. A substitution using 0u u uD = - can also be done here.
The LP formulation restricts the control variables to move only within their
respective limits, but it does not yet constrain the OPF to optimize cost within the
limits of transmission flows and load bus voltages. For this a new constraint is added
in the LP like MVA flow on line nm is constraint to fall within an upper limit:
MVA flownm ≤ MVA flownmmax
This constraint is modeled by forming a Taylor’s series expansion and only retaining
the linear terms:
0 maxnm nm nm nm
uMVAflow MVAflow MVAflow u MVAflow
u¶æ ö= + D £ç ÷¶è ø
å
we can substitute 0u u uD = - so we get:
- 18 -
max
0 0
nm nm fu
f nm nmu
MVAflow u MVAflow Ku
where
K MVAflow MVAflow uu
¶æ ö £ -ç ÷¶è ø
¶= +
¶
å
å
Other constraints such as voltage magnitude limits, branch MW limits, etc., can be
added in a similar manner. We can add as many constraints as necessary to constraint
the power system to remain within its prescribed limits. The derivatives of Ploss and
MVA flownm are obtained from the linear sensitivity coefficient calculations [3,21].
3.3.1 Algorithm
Step 1- Input bus data, line data, cost functions, maximum and minimum values of
control parameters etc.
Step 2- Solve power flow equations using any method like Gauss Seidel Method,
Newton Raphson, Fast decoupled etc.
Step 3- Linearize the objective function.
Step 4- Obtain linearized constraint sensitivity coefficients.
Step 5- Get the solution using Linear Programming incorporating equality and
inequality constraints of control variables.
Step 6- Test Convergence of solution.
Step 7- If it converges then there is no movement of control variables and we get the
solution.
Step 8- If it does not converge, then adjust the control variables and go to step 2.
The results of the Linear Programming based Optimal Power Flow are given in
Chapter 4.
- 19 -
Chapter 4
Selection for the Proper Location of Series Capacitor
4.1 IntroductionMany researches were made on optimal site selections for placement of
compensating devices in power systems by considering different criteria. In this
chapter, various parameters will be considered for their optimal allocations like
sensitivity analysis, line flow index etc..
The best location, appropriate size and setting of compensating devices
are important in the deregulated electricity markets. On one hand, there is
considerable risk in the investment of these devices because of their high cost and the
uncertain market transactions. There is a need to maximize the benefit from
investment in compensating devices. On the other hand, the location and setting of
these devices have direct and discriminatory impact on the market participants.
Implementation of these devices can distort the prices at buses, directly affecting
some generators and loads. Therefore, the location and setting of these devices require
crucial consideration.
Capacitor banks are widely used in distribution systems for reactive power
compensation, in order to achieve power and energy loss reduction, system capacity
release and acceptable voltage profiles. The extent of these benefits depends on the
location, size, type and number of capacitors (sources of reactive power) placed in the
system. Hence an optimal solution for placement and sizing of capacitors in a
distribution system is a very important aspect of power system analysis.
It is known that power flow through an AC transmission line is a function of
line impedance, the magnitude and the phase angles between the sending and
receiving end voltages. Variable series capacitors can be utilized to change the power
flow by changing the parameters of the networks. Thus the power transfer capabilities
will be increased and the transmission investment cost will be reduced [29].
Various parameters like sensitivity coefficients, line flow index etc. are
calculated and analyzed to select the proper location of the device. These parameters
are discussed below.
- 20 -
4.2 Various Parameters
4.2.1 Sensitivity Coefficients
Sensitivity coefficients give an indication of the change in one system quantity
(e.g. MW flow, MVA flow, bus voltage etc.) as another quantity is varied (e.g.
generator MW output, transformer tap position etc.). As the adjustable variable is
changed, we assume that the power system reacts so as to keep all of the power flow
equations solved. ij kMVAflow MWgen¶ ¶ shows the sensitivity of the flow (MVA) on
line (i to j) with respect to the power generated at bus k. In this work we calculate the
change in active power (reactive power) flow with respect to the voltage at the bus
and power angle. Eight parameters ij iP V¶ ¶ , ij jP V¶ ¶ ,
, , , , ,ij i ij j ij i ij j ij i ij jP P Q V Q V Q Qd d d d¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ are calculated [6,22].
As we know,
* Flow,Sij i ij ik ijLine V I P jQ= ´ = +
Sij is the line flow between line i and j.
2 2 2 22 2
2 2 2 22 2
cos cos cos cos cos cos sin
sin cos sin cos sin
sin sin cos cos sin sin
ij ij ijij i i i j i j i i
iji k i j i n i i
ij iji j i j i j i j
Y Y Ythen P V VV V
a a aY Bc BcVV V Va a a
Y YVV VV
a a
d q q d d q d
q d d d d
q d d q d d
= - +
+ + +
- -
2 2 2 22 2and Q cos sin sin cos cos sin sin
cos cos sin cos sin cos sin sin sin
ij ij ijij i i i j i j i i
ij ij iji k i j i j i j i j i j
Y Y YV VV V
a a aY Y Y
VV VV VVa a a
d q q d d q d
q d d q d d q d d
= - + -
+ - +
- 21 -
then various coefficients are calculated from above equations. These are given below-
( )
( )
( )
( )
2 22 cos cos 2
cos
sin
sin
ij ij iji j j i i
i
ij iji i j
j
ij iji j i j
i
ij iji j j i
j
P Y Y BcV V VV a a aP Y
VV aP Y
VVa
P YVV
a
q d d q
d d q
d d qd
d d qd
¶= + - + +
¶
¶= - - -
¶
¶= - -
¶
¶= - +
¶
( )
( )
( )
( )
22 sin cos 2 sin
sin
cos
cos
ij ij iji i j j i
i
ij iji j i
j
ij iji j j i
i
ij iji j i j
j
Q Y YV V
V a aQ Y
VV aQ Y
VVa
Q YVV
a
q d d d q
d d q
d d qd
d d qd
¶= - + - +
¶
¶= - +
¶
¶= - - +
¶
¶= - -
¶
Where
Sij is the Apparent Power Flow from ith bus to jth bus.
Pij is Active Power Flow from ith bus to jth bus.
Qij is Reactive Power Flow from ith bus to jth bus.
Vi is Voltage at bus i.
Vj is the Voltage at bus j.
Yij is the Admittance between bus i and j.
a is the Tap Changing Ratio.
θ is the Power factor angle.
δi is the Power Angle of bus i.
Bc is Line Charging Admittance value.
These sensitivity coefficients are calculated for IEEE 30 bus system. Its Bus data and
Line data are given in Appendix I. There are 41 lines in the system. Sensitivity
coefficients are calculated for these lines. Change in Active Power is more sensitive to
power angle and change in Reactive Power is more sensitive to voltage of the bus.
Sensitivity of the line is high if its coefficient values are high.
- 22 -
4.2.2 Line Flow Index
It is the ratio of the line flow between any two buses of the system to the total
connected load of the system. Total connected load is constant for a particular system
so it is like a base load value. This index is calculated for IEEE 30 bus test system.
More is value of index more is the line flow between the buses.
4.2.3 Voltage Drop
Voltage drops are calculated by multiplying line flow between two buses to
the impedance of the line. Resistive Drop and Reactive Drop are calculated.
( ) ( )Resistive Drop = -Reactive Drop =
D P jQ R jXPR QXQR XP
= + ´ +
+
These values are calculated for all lines of IEEE 30 bus test system.
Above parameters are calculated and analyzed to select the proper location of Series
Capacitor.
4.3 Results and DiscussionsVarious parameters discussed previously are calculated for IEEE 30 bus test
system. Its code is developed in MATLAB. Its Bus data and Line data is given in
Appendix I. Values of these parameters are given in table below-
Line No. i to j ∂P(i,j)/∂V(i) (S1) ∂P(i,j)/∂V(j)(S2) ∂P(i,j)/∂δ(i)(S3) ∂P(i,j)/∂δ(j)(S4)1 1--2 14.943302 -4.036904 17.399608 -17.3996082 1--3 3.245266 -0.64541 5.636709 -5.6367093 2--4 4.951505 -1.438228 5.604189 -5.6041894 3--4 24.693873 -7.840675 25.075666 -25.0756665 2--5 3.413907 -0.702166 5.452078 -5.4520786 2--6 4.760927 -1.2922 5.527199 -5.5271997 4--6 19.147796 -6.019656 23.521307 -23.5213078 5--7 9.022821 -3.076588 7.50257 -7.502579 6--7 10.663582 -3.377903 11.429276 -11.429276
10 6--8 19.060579 -6.204811 23.057083 -23.05708311 6--9 0.192142 -0.188239 -5.035058 5.03505812 6--10 0.133718 -0.131695 -1.97282 1.9728213 9--11 -0.095501 0.091634 -5.442643 5.44264314 9--10 0.284179 -0.285685 -9.81987 9.8198715 4--12 0.312163 -0.30645 -4.11942 4.1194216 12--13 -0.119292 0.114511 -8.432917 8.43291717 12--14 5.023804 -1.605428 3.572892 -3.57289218 12--15 10.152652 -3.236238 6.846911 -6.84691119 12--16 6.443701 -2.065988 4.629608 -4.629608
- 23 -
20 14--15 2.278728 -0.756561 1.772042 1.77204221 16--17 5.75749 -1.90115 4.680064 -4.68006422 15--18 5.387097 -1.787385 3.823651 -3.82365123 18--19 9.359667 -3.107307 6.43533 -6.4353324 19--20 17.987637 -6.013466 12.166565 -12.16656525 10--20 5.464152 -1.792051 4.241141 -4.24114126 10--17 12.254373 -4.075507 11.034744 -11.03474427 10--21 15.71812 -5.20381 11.696999 -11.69699928 10--22 8.074631 -2.675546 5.755198 -5.75519829 21--22 51.543916 -17.187503 35.783183 -35.78318330 15--23 5.882231 -1.966045 4.102346 -4.10234631 22-24 7.752496 -2.582372 4.106407 -4.10640732 23--24 4.436986 -1.475727 3.072464 -3.07246433 24--25 3.988458 -1.329842 2.352333 -2.35233334 25--26 3.683307 -1.225818 1.862429 -1.86242935 25--27 6.071174 -2.030731 3.943224 -3.94322436 28--27 -0.179746 0.182603 -2.963223 2.96322337 27--29 3.024075 -0.986824 1.989111 -1.98911138 27--30 2.06096 -0.661046 1.363203 -1.36320339 29--30 2.735048 -0.897373 1.76226 -1.7622640 8--28 4.422012 -1.477033 4.721892 -4.72189241 6--28 13.192289 -4.301784 16.093802 -16.093802
Line No. i to j ∂Q(i,j)/∂V(i)(S5) ∂Q(i,j)/∂V(j)(S6) ∂Q(i,j)/∂δ(i)(S7) ∂Q(i,j)/∂δ(j)(S8)1 1--2 16.287071 -16.843764 -4.170122 4.1701222 1--3 5.333349 -5.469934 -0.665088 0.6650883 2--4 5.150333 -5.464476 -1.475 1.4754 3--4 22.727992 -24.450524 -8.041142 8.0411425 2--5 5.147529 -5.420638 -0.706238 0.7062386 2--6 5.060247 -5.412824 -1.319505 1.3195057 4--6 20.871084 -23.034576 -6.146854 6.1468548 5--7 6.215386 -7.4498 -3.098381 3.0983819 6--7 10.020679 -11.348886 -3.40183 3.4018310 6--8 19.770652 -22.538693 -6.347521 6.34752111 6--9 -4.177604 4.830682 -0.196203 0.19620312 6--10 -1.661629 1.902775 -0.136543 0.13654313 9--11 -3.929515 5.010258 0.099542 -0.09954214 9--10 -7.882884 9.471218 -0.296202 0.29620215 4--12 -3.555164 3.943213 -0.320144 0.32014416 12--13 -6.431119 7.748706 0.124622 -0.12462217 12--14 3.023466 -3.466948 -1.654487 1.65448718 12--15 5.826268 -6.675731 -3.319222 3.31922219 12--16 3.90218 -4.477244 -2.136295 2.13629520 14--15 1.343652 -1.72774 -0.775961 0.77596121 16--17 3.541108 -4.541165 -1.959299 1.95929922 15--18 2.799983 -3.75864 -1.8183 1.818323 18--19 4.726749 -6.336947 -3.155549 3.15554924 19--20 8.81281 -11.927379 -6.134058 6.13405825 10--20 3.238585 -4.157763 -1.827988 1.827988
- 24 -
26 10--17 8.331014 -10.707244 -4.200163 4.20016327 10--21 8.912031 -11.424429 -5.327965 5.32796528 10--22 4.381253 -5.618898 -2.740447 2.74044729 21--22 26.534916 -34.935736 -17.604426 17.60442630 15--23 3.031624 -4.039369 -1.996697 1.99669731 22-24 3.120629 -4.062974 -2.609978 2.60997832 23--24 2.306582 -3.039966 -1.491503 1.49150333 24--25 1.720881 -2.308662 -1.354998 1.35499834 25--26 1.423233 -1.860632 -1.227002 1.22700235 25--27 2.857449 -3.818577 -2.097018 2.09701836 28--27 -2.294175 2.915159 0.185613 -0.18561337 27--29 1.507691 -1.963661 -0.999613 0.99961338 27--30 1.042136 -1.361039 -0.662097 0.66209739 29--30 1.274834 -1.759463 -0.8988 0.898840 8--28 4.074634 -4.645303 -1.501386 1.50138641 6--28 13.990024 -15.83276 -4.372709 4.372709
Line No. i to j Line flow/conn.load R drop X drop S1 1—2 0.505 0.040776 0.088008 37.8002 1—3 0.223 0.039466 0.12893 11.56473 2—4 0.111 0.033153 0.055493 12.25304 3—4 0.214 0.012237 0.023858 56.32545 2—5 0.22 0.034508 0.12849 11.33476 2—6 0.153 0.043265 0.07951 11.99267 4—6 0.193 0.0095736 0.024067 50.50888 5—7 0.038 -0.0077442 -0.011694 17.79849 6—7 0.111 0.0043044 0.029588 25.2732
10 6—8 0.075 0.0071279 0.0075221 49.442011 6—9 0.087 0.036026 0.040802 9.572912 6—10 0.046 -0.027811 0.075912 3.773313 9—11 0.073 0.040615 -0.020783 9.9912814 9—10 0.097 -0.0061619 0.032578 18.575215 4—12 0.113 0.029387 0.081973 7.9071916 12—13 0.116 0.045468 -0.0168 15.610517 12—14 0.024 0.0030874 0.021483 8.943318 12—15 0.061 0.0040386 0.027797 17.545819 12—16 0.024 0.0026238 0.016619 11.53520 14—15 0.004 -0.00055164 0.0063548 4.247321 16—17 0.012 0.0026133 0.0075228 11.018522 15—18 0.019 0.0041562 0.014188 9.488723 18—19 0.009 0.0016884 0.0035886 16.201824 19—20 0.024 -0.000014379 -0.0057334 30.837425 10—20 0.032 -0.00048304 0.022949 10.173526 10—17 0.025 -0.0030421 0.0062513 25.092727 10—21 0.06 -0.0025662 0.015434 28.539328 10—22 0.029 -0.0020928 0.014948 14.296529 21—22 0.007 -0.000050541 -0.00055469 89.7635
- 25 -
30 15—23 0.018 -0.00070198 0.012436 10.276431 22-24 0.022 -0.0011534 0.014627 11.847532 23—24 0.006 -0.00080222 0.005669 7.7301733 24—25 0.007 0.0019209 -0.0084798 6.3801734 25—26 0.013 0.000024592 0.019489 5.5231535 25—27 0.021 0.0024794 -0.01524 10.179536 28—27 0.064 0.027091 -0.068503 5.6086837 27—29 0.02 0.0066884 0.029356 5.114638 27—30 0.023 0.01271 0.048033 3.4951939 29—30 0.012 0.0061436 0.018236 4.5643740 8—28 0.017 -0.0009573 0.0024572 10.4411741 6—28 0.058 0.0015835 0.010964 34.571679
where
2 2 2 21 2 7 8........S S S S S= + + + +
Depending on these parameters the optimal location of the compensating device is
selected.
Based on the values of S, Line Flow Index and Reactive Drop optimal location is
found out. Location is selected where value of sensitivity is high and Line Flow is
noticeable. Reactive Drop value is also considered. IEEE 30 Bus Test System is used.
Its data is given in Appendix I. Based on the above parameters following lines are
selected for compensating device-
· Line No. 4 between buses 3-4
· Line No. 7 between buses 4-6
· Line No. 10 between buses 6-8
· Line No. 13 between buses 9-11
· Line No. 16 between buses 12-13
· Line No. 29 between buses 21-22
· Line No. 34 between buses 25-26
A fixed capacitor having capacitive reactance of 0.02 ohm is placed in these lines.
The reactive power compensated by the capacitor is given below:
Sr. No. Line No. Compensated power (MVar)
1 4 2.0778
2 7 2.7555
3 10 5.5201
4 13 3.2745
- 26 -
5 16 2.9544
6 29 0.15355
7 34 0.0033792
At Line no. 34 compensated power is very low inspite of high value of Sensitivity
Index; this is due to the very low value of Line Flow Index.
After placing the Series Capacitor system is optimized by using Linear
Programming with only the power balance equation in the LP constraint set.
Generation limits are incorporated. The unit cost functions are broken into three
straight line segments. The break points taken for this problem are given below:
Unit Break point 1
(Unit min.)
Break point 2 Break point 3 Break point 4
(Unit max.)
1 50 100 150 200
2 20 40 60 80
3 15 30 40 50
4 10 15 25 35
5 10 17 22 30
6 12 20 30 40
After running the AC power flow,
Line losses are 13.221 MW and 18.265 MVar.
Total cost for dispatch is 833.27$/hr.
By applying LPOPF on these selected lines, the Line losses and operating cost
obtained are:
Sr. No. Line No. Losses (MW) Losses (MVar) Cost ($/hr.)
1 4 11.407 12.516 808.37
2 7 11.401 12.528 808.35
3 10 11.409 12.631 808.39
4 13 11.410 12.633 808.39
5 16 11.408 12.667 808.38
6 29 11.406 12.522 808.38
7 34 11.406 12.521 808.38
- 27 -
Chapter 5
Conclusion and Future Work
5.1 ConclusionThis thesis deals with the problem of allocation of compensating device. The
aim is to find the location of series capacitor in given power system such that we get
maximum reactive power compensation. Various parameters like sensitivity
coefficients, line flow index etc. has been calculated. By analyzing these parameters
optimal location of series capacitor is found out. Firstly, LPOPF is implemented on
parent system. Then, a known value of series capacitor is placed in the selected
location or line. It changes the system parameters while running the load flow.
LPOPF is obtained for the compensated system. The pre-compensated system results
are then compared with post-compensated results. IEEE 30 bus test system is
considered for the abovesaid problems. It has been noted that
· At optimal location reactive compensated power is maximum.
· System losses has been decreased.
· Operating cost is also minimum.
5.2 Future WorkThe following points are recommended for future extension of work-
· Solve the problem by replacing other compensating devices like Flexible AC
Transmission Systems (FACTS) devices instead of Series capacitor. FACTS
device provide both series and shunt compensation.
· Optimal location can be found out by artificial techniques like Evolutionary
Programming or other Heuristic techniques.
- 28 -
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- 31 -
Appendix I
IEEE 30 BUS TEST SYSTEMBus data =
Busno.
Buscode
Vol.mag.
Angle(deg.)
LoadMW
LoadMVar
Gen.MW
Gen.MVar
Gen.Qmin
Gen.Qmax.
Inj.MVar
1 1 1.05 0 0.0 0.0 50 0 -20 150 02 2 1.033 0 21.70 12.7 20 0 -30 60 03 0 1.0 0 2.4 1.2 0 0 0 0 04 0 1.0 0 7.6 1.6 0 0 0 0 05 2 1.0058 0 94.2 19.0 15 0 -15 60 06 0 1.0 0 0.0 0.0 0 0 0 0 07 0 1.0 0 22.8 10.9 0 0 0 0 08 2 1.023 0 30.0 30.0 10 0 -15 50 09 0 1.0 0 0.0 0.0 0 0 0 0 0
10 0 1.0 0 5.8 2.0 0 0 0 0 1911 2 1.0913 0 0.0 0.0 10 0 -10 -40 012 0 1.0 0 11.2 7.5 0 0 0 0 013 2 1.0883 0 0.0 0.0 12 0 -15 45 014 0 1.0 0 6.2 1.6 0 0 0 0 015 0 1.0 0 8.2 2.5 0 0 0 0 016 0 1.0 0 3.5 1.8 0 0 0 0 017 0 1.0 0 9.0 5.8 0 0 0 0 018 0 1.0 0 3.2 0.9 0 0 0 0 019 0 1.0 0 9.5 3.4 0 0 0 0 020 0 1.0 0 2.2 0.7 0 0 0 0 021 0 1.0 0 17.5 11.2 0 0 0 0 022 0 1.0 0 0.0 0.0 0 0 0 0 023 0 1.0 0 3.2 1.6 0 0 0 0 024 0 1.0 0 8.7 6.7 0 0 0 0 025 0 1.0 0 0.0 0.0 0 0 0 0 026 0 1.0 0 3.5 2.3 0 0 0 0 027 0 1.0 0 0.0 0.0 0 0 0 0 028 0 1.0 0 0.0 0.0 0 0 0 0 029 0 1.0 0 2.4 0.9 0 0 0 0 030 0 1.0 0 10.6 1.9 0 0 0 0 0
- 32 -
Line data =
Bus from Bus to R (pu) X (pu) 0.5 B (pu) Tap Ratio MW flowlimit
1 2 0.0192 0.0575 0.0264 1 1.31 3 0.0452 0.1852 0.02040 1 1.32 4 0.0570 0.1737 0.0184 1 .653 4 0.0132 0.0379 0.00420 1 1.32 5 0.0472 0.183 0.02090 1 1.32 6 0.0581 0.1763 0.01870 1 .654 6 0.0119 0.0414 0.00450 1 .95 7 0.0460 0.1160 0.01020 1 1.36 7 0.0267 0.0820 0.00850 1 1.36 8 0.0120 0.0420 0.00450 1 .326 9 0.0 0.2080 0.0 0.978 .656 10 0.0 0.5560 0.0 0.969 .329 11 0.0 0.2080 0.0 1 .659 10 0.0 0.1100 0.0 1 .654 12 0.0 0.2560 0.0 0.932 .65
12 13 0.0 0.1400 0.0 1 .6512 14 0.1231 0.2559 0.0 1 .3212 15 0.0662 0.1304 0.0 1 .3212 16 0.0945 0.1987 0.0 1 .3214 15 0.2210 0.1997 0.0 1 .1616 17 0.0824 0.1923 0.0 1 .1615 18 0.1073 0.2185 0.0 1 .1618 19 0.0639 0.1292 0.0 1 .1619 20 0.0340 0.0680 0.0 1 .3210 20 0.0936 0.2090 0.0 1 .3210 17 0.0324 0.0845 0.0 1 .3210 21 0.0348 0.0749 0.0 1 .3210 22 0.0727 0.1499 0.0 1 .3221 22 0.0116 0.0236 0.0 1 .3215 23 0.1000 0.2020 0.0 1 .1622 24 0.1150 0.1790 0.0 1 .1623 24 0.1320 0.27 0.0 1 .1624 25 0.1885 0.3292 0.0 1 .1625 26 0.2544 0.38 0.0 1 .1625 27 0.1093 0.2087 0.0 1 .1628 27 0.0 0.3960 0.0 0.968 .6527 29 0.2198 0.4153 0.0 1 .1627 30 0.3202 0.6027 0.0 1 .1629 30 0.2399 0.4533 0.0 1 .168 28 0.0636 0.2 0.0214 1 .326 28 0.0169 0.0599 0.065 1 .32
- 33 -
Cost Coefficients=
c b a0 2 0.003750 1.75 0.017500 1.00 0.062500 3.25 0.008340 3 0.025000 3 0.02500
Limits=
MW (min) MW (max) MVar (min) MVar (max)
50 200 -20 150
20 80 -20 60
15 50 -15 60
10 35 -15 50
10 30 -10 40
12 40 -15 45